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## G = D5.D8order 160 = 25·5

### The non-split extension by D5 of D8 acting via D8/C8=C2

Aliases: C81F5, C401C4, D5.1D8, D10.9D4, D5.1Q16, Dic5.3Q8, C5⋊(C2.D8), C52C86C4, C4⋊F5.4C2, C4.9(C2×F5), C20.9(C2×C4), (C8×D5).3C2, C2.5(C4⋊F5), C10.2(C4⋊C4), (C4×D5).26C22, SmallGroup(160,69)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D5.D8
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4⋊F5 — D5.D8
 Lower central C5 — C10 — C20 — D5.D8
 Upper central C1 — C2 — C4 — C8

Generators and relations for D5.D8
G = < a,b,c,d | a5=b2=c8=1, d2=a-1b, bab=a-1, ac=ca, dad-1=a3, bc=cb, dbd-1=a2b, dcd-1=c-1 >

Character table of D5.D8

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5 8A 8B 8C 8D 10 20A 20B 40A 40B 40C 40D size 1 1 5 5 2 10 20 20 20 20 4 2 2 10 10 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 -1 1 -1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 -1 -1 1 -1 i i -i -i 1 1 1 -1 -1 1 1 1 1 1 1 1 linear of order 4 ρ6 1 1 -1 -1 1 -1 -i i i -i 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ7 1 1 -1 -1 1 -1 -i -i i i 1 1 1 -1 -1 1 1 1 1 1 1 1 linear of order 4 ρ8 1 1 -1 -1 1 -1 i -i -i i 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ9 2 2 2 2 -2 -2 0 0 0 0 2 0 0 0 0 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 2 -2 0 0 0 0 0 0 2 -√2 √2 √2 -√2 -2 0 0 √2 √2 -√2 -√2 orthogonal lifted from D8 ρ11 2 -2 2 -2 0 0 0 0 0 0 2 √2 -√2 -√2 √2 -2 0 0 -√2 -√2 √2 √2 orthogonal lifted from D8 ρ12 2 2 -2 -2 -2 2 0 0 0 0 2 0 0 0 0 2 -2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ13 2 -2 -2 2 0 0 0 0 0 0 2 -√2 √2 -√2 √2 -2 0 0 √2 √2 -√2 -√2 symplectic lifted from Q16, Schur index 2 ρ14 2 -2 -2 2 0 0 0 0 0 0 2 √2 -√2 √2 -√2 -2 0 0 -√2 -√2 √2 √2 symplectic lifted from Q16, Schur index 2 ρ15 4 4 0 0 4 0 0 0 0 0 -1 -4 -4 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from C2×F5 ρ16 4 4 0 0 4 0 0 0 0 0 -1 4 4 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ17 4 4 0 0 -4 0 0 0 0 0 -1 0 0 0 0 -1 1 1 -√-5 √-5 √-5 -√-5 complex lifted from C4⋊F5 ρ18 4 4 0 0 -4 0 0 0 0 0 -1 0 0 0 0 -1 1 1 √-5 -√-5 -√-5 √-5 complex lifted from C4⋊F5 ρ19 4 -4 0 0 0 0 0 0 0 0 -1 -2√2 2√2 0 0 1 √-5 -√-5 ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 complex faithful ρ20 4 -4 0 0 0 0 0 0 0 0 -1 2√2 -2√2 0 0 1 -√-5 √-5 ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 complex faithful ρ21 4 -4 0 0 0 0 0 0 0 0 -1 2√2 -2√2 0 0 1 √-5 -√-5 ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 complex faithful ρ22 4 -4 0 0 0 0 0 0 0 0 -1 -2√2 2√2 0 0 1 -√-5 √-5 ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 complex faithful

Smallest permutation representation of D5.D8
On 40 points
Generators in S40
```(1 26 12 24 35)(2 27 13 17 36)(3 28 14 18 37)(4 29 15 19 38)(5 30 16 20 39)(6 31 9 21 40)(7 32 10 22 33)(8 25 11 23 34)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 33)(8 34)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)
(1 8)(2 7)(3 6)(4 5)(9 37 21 28)(10 36 22 27)(11 35 23 26)(12 34 24 25)(13 33 17 32)(14 40 18 31)(15 39 19 30)(16 38 20 29)```

`G:=sub<Sym(40)| (1,26,12,24,35)(2,27,13,17,36)(3,28,14,18,37)(4,29,15,19,38)(5,30,16,20,39)(6,31,9,21,40)(7,32,10,22,33)(8,25,11,23,34), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40), (1,8)(2,7)(3,6)(4,5)(9,37,21,28)(10,36,22,27)(11,35,23,26)(12,34,24,25)(13,33,17,32)(14,40,18,31)(15,39,19,30)(16,38,20,29)>;`

`G:=Group( (1,26,12,24,35)(2,27,13,17,36)(3,28,14,18,37)(4,29,15,19,38)(5,30,16,20,39)(6,31,9,21,40)(7,32,10,22,33)(8,25,11,23,34), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40), (1,8)(2,7)(3,6)(4,5)(9,37,21,28)(10,36,22,27)(11,35,23,26)(12,34,24,25)(13,33,17,32)(14,40,18,31)(15,39,19,30)(16,38,20,29) );`

`G=PermutationGroup([[(1,26,12,24,35),(2,27,13,17,36),(3,28,14,18,37),(4,29,15,19,38),(5,30,16,20,39),(6,31,9,21,40),(7,32,10,22,33),(8,25,11,23,34)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,33),(8,34),(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(23,25),(24,26)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40)], [(1,8),(2,7),(3,6),(4,5),(9,37,21,28),(10,36,22,27),(11,35,23,26),(12,34,24,25),(13,33,17,32),(14,40,18,31),(15,39,19,30),(16,38,20,29)]])`

D5.D8 is a maximal subgroup of
C802C4  C803C4  D5.D16  D5.Q32  (C2×C8)⋊6F5  M4(2)⋊1F5  D8×F5  SD16⋊F5  Q16×F5  Dic5.4Dic6  D5.D24
D5.D8 is a maximal quotient of
C802C4  C803C4  C16.F5  C80.2C4  C401C8  Dic5.13D8  D10.10D8  Dic5.4Dic6  D5.D24

Matrix representation of D5.D8 in GL4(𝔽7) generated by

 3 1 3 3 0 1 6 3 4 4 2 2 1 5 6 0
,
 6 0 4 2 0 6 1 4 0 3 3 1 0 1 3 6
,
 3 3 0 2 1 3 1 1 2 2 6 3 3 4 2 3
,
 4 1 3 3 6 2 6 6 1 2 1 2 0 2 1 0
`G:=sub<GL(4,GF(7))| [3,0,4,1,1,1,4,5,3,6,2,6,3,3,2,0],[6,0,0,0,0,6,3,1,4,1,3,3,2,4,1,6],[3,1,2,3,3,3,2,4,0,1,6,2,2,1,3,3],[4,6,1,0,1,2,2,2,3,6,1,1,3,6,2,0] >;`

D5.D8 in GAP, Magma, Sage, TeX

`D_5.D_8`
`% in TeX`

`G:=Group("D5.D8");`
`// GroupNames label`

`G:=SmallGroup(160,69);`
`// by ID`

`G=gap.SmallGroup(160,69);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,151,579,69,2309,1169]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^2=c^8=1,d^2=a^-1*b,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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